If a signal is discrete in time, it has a Fourier transform that is both continuous and periodic in frequency. We can see this by interchanging the time and frequency parameters: the discrete time sequence could be viewed as the Fourier series of the continuous transform.
More importantly, if the discrete time sequence is a sampled version of a continuous time signal and the sampling rate is greater than twice the highest frequency of the continuous signal, then the Fourier transform of the sampled signal will be a periodic version of the Fourier transform of the original, continuous waveform. All of the information in the continuous waveform can be derived from the sampled waveform.
The Fourier transform of a discrete, non-periodic signal is called the Discrete Time Fourier Transform or DTFT.
A discrete, non-periodic signal with a sample period of T has the following Fourier transform: